A Prediction of Aerial Shell and Comet Trajectories Using SHELLCALC©

John Harradine and Tom Smith

Abstract: This paper describes a model for predicting the path of aerial shells and Roman candle comets. This model, incorporated in a Microsoft® Excel-based freeware program, SHELLCALC©, predicts the trajectoryof these fireworks using point mass equations for range and height. These equations are modifi ed to take into consideration mortar/candle angle, launch altitude above sea level, wind speed and direction, comet consumption, air density and terrain, and incorporate an approximation of shell drift through tumbling motion and mortar balloting. The graphical output from the model also incorporates typical shell burst diameters.

Keywords: Shells, comets, ballistics, trajectory, fall-out

NOTE – A new version of the Shellcalc© program is now available at a dedicated website – please click here



Ref: JPyro, Issue 22, 2005, pp9-15

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9 Responses to “A Prediction of Aerial Shell and Comet Trajectories Using SHELLCALC©”

  • toms:

    Shellcalc has now been adopted by the British Pyrotechnists Association for site and product specific Risk assessments of firework displays.

  • VStrauch:

    Shellcalc is now used on my Suggestion by Slovenian Firework display Operators for Hazard or Risk Assessment connected to Magnus Effect, angle of Mortar and Wind.

    Dr Strauch from Ljubljana

  • toms:

    Also see a follow-up paper at


    Tom Smith

  • […] A new version of the Shellcalc© program is now available.  The original paper describing the program is available here. […]

  • DannyB:

    Would be nice to be able to angle tubes in the Y axis as well. Also should be fairly easy to implement tick boxes for muzzle break and 99s delay. What I want to know on most shows is where is it going to go when it goes right, what if it muzzle breaks and what if it bursts on the ground. Being able to get to that quickly would be useful.

  • toms:

    The Y-axis issue isn’t really necessary – leaning a mortar over in the Y-axis is only the same as leaning it over in the X-axis and rotating everything by 90 degrees. Leaning it over to X and Y is the same as rotating the model by 45 degrees – etc etc. I think it is more flexible this way really – but importantly what Shellcalc should be used for is prediction of most-credible and worst-case scenarios and in that respect treat the wind as relative – we usually model at 45 degree increments on a large display site and if there are 3 firing angles (angled one way, straight up and angled the other way) that would mean 9 x 3 = 27 models – but in practice it is a lot less if you think about the possibilities logically.

    Bursting at the muzzle is rarely the major determinant of a “safety” distance (or at least it shouldn’t be) but setting the delay time to zero achieves that model, as does setting it to 99 to get ground burst after flight. Entering 0 or 99 isn’t really significantly more keypresses or clicks and again gives flexibility – but I will think about it.

  • admin:

    Danny. With the new Shellcalc Pro there is the option of modelling multiple effects at once – and now there is a need for a y-axis rotation to allow firing from different points to be oriented in different directions. What I’ve done is to add a new parameter – rotation – so you can set a firing angle, and a rotation for each effect.

    At present this is only in the development version – the existing ShellCalc Pro allows 3 simultaneous effects (and a load of other things) but they are all fired in the same plane. The next public version will allow rotation

  • dtkhanh:

    Help me, I live in Viêt Nam
    Can you send calculator Hmax and Lmax, Please 😀